The Timpson Lyceum · First-Order Predicate Logic (2024)

Axioms of Weak Preference & Derivations with Universal Quantifiers - Lecture 6 (Logic)

Axioms of Weak Preference & Derivations with Universal Quantifiers – Lecture 6

Overview

This lecture introduces the two derivation rules that let us reason about the universal quantifier $\forall$ inside formal proofs, and then puts them to work in our first complete axiomatic system. After a recap of the grammar of first-order logic (terms, operators, predicates, relations, and quantifiers), we add Universal Specification (US) and Universal Generalization (UG) to our existing toolkit of derivation rules. We learn the four-step proof flow — symbolize, drop quantifiers, reason, re-quantify — and why a "flagged" variable can sabotage the last step. We then build the axiomatic system of weak preference, with two axioms and two definitions, and prove several of its theorems. The payoff is seeing how a single undefined relation, constrained only by its axioms, generates a rich body of provable consequences.

Key Concepts

Grammar recap — the four symbol categories. Every symbol we write (besides parentheses) is one of: sentential connectives ( $\lnot, \land, \lor, \rightarrow, \leftrightarrow$ ); terms (constants, variables, and operators); predicates; or quantifiers.

Term — anything that evaluates to a single object when its variables are replaced by constants (e.g. $1$ , $x$ , $19x - 7$ ).
Operator — a term-valued function: it takes terms and returns a term (e.g. $+$ , $-$ ).
Predicate — a proposition-valued function: give it its terms and it returns something true or false. A predicate of one term is a property; a predicate of two terms is a relation (e.g. $<$ ). Order matters: $1 < 2$ is true, $2 < 1$ is false.
Quantifiers — the universal $\forall$ ("for all / for each / for every") and the existential $\exists$ ("there exists at least one"). $\exists!$ means "there exists exactly one," noted but not used.
Distributing negation across quantifiers flips them: $\lnot \forall x\, P(x)$ becomes $\exists x\, \lnot P(x)$ , and $\lnot \exists x\, P(x)$ becomes $\forall x\, \lnot P(x)$ .
Order of quantifiers matters. $\forall x\, \exists y\,(y < x)$ ("every number has a smaller one") is not the same as $\exists y\, \forall x\,(y < x)$ ("some number is below every number").

Rule US — Universal Specification (removing $\forall$ ). If a formula $S$ results from a formula $R$ by substituting a term $t$ for every occurrence of the variable $x$ in $R$ , then $S$ is derivable from $\forall x\, R$ . In words: if something holds for every object, it holds for any specific term you choose — a constant, another variable, or any compound term. The only requirement is consistency: replace all occurrences of $x$ with the same term.

Rule UG — Universal Generalization (restoring $\forall$ ). From a formula $S$ you may derive $\forall x\, S$ , provided $x$ is not flagged in $S$ .

Flagged variable. A variable that is both free and introduced in a premise. You may not universally generalize on a flagged variable, nor on anything derived from a premise that introduced it. Without this restriction you could "prove" nonsense — e.g. that if anything is human then everything is human — by generalizing on a variable that was only assumed.

The four-step proof flow with quantifiers.

Symbolize the premises.
Drop the quantifiers using US.
Apply the ordinary rules of derivation (P, CP, T, RAA).
Re-attach the quantifiers using UG.

Existing derivation rules (carried over). Rule P (introduce a premise), rule CP (conditional premise, later discharged), rule T (use any tautology / rule of logic), rule RAA (reductio ad absurdum: if a premise yields a contradiction, that premise is false).

Anatomy of an axiomatic system. Four components: undefined terms (the predicates, operators, or constants taken as primitive), axioms (initial assumptions that give the undefined terms meaning), definitions (new terms built from existing ones — you cannot define everything, or you get circularity or infinite regress), and theorems (the logical consequences of the axioms).

The system of weak preference. The single undefined term is the relation $Q$ , read $xQy$ as "weakly prefer $x$ to $y$ ." $Q$ means nothing more than what its axioms say.

Axiom A1 (transitivity): $\forall x\, \forall y\, \forall z\,\big((xQy \land yQz) \rightarrow xQz\big)$ .
Axiom A2 (totality): $\forall x\, \forall y\,(xQy \lor yQx)$ — for any two objects you weakly prefer one to the other.
Definition D1 (indifference): $x\,I\,y \;\leftrightarrow\; (xQy \land yQx)$ .
Definition D2 (strong preference): $x\,P\,y \;\leftrightarrow\; \lnot\, yQx$ . (The " $xQy$ " half is omitted because A2 already supplies it — a more minimal definition.)

Indifference is proved to be reflexive, symmetric, and transitive, i.e. an equivalence relation. Axioms/definitions are referenced directly as lines A1, A2, D1, D2 without rewriting them.

Worked Example

Theorem (symmetry of indifference): $\forall x\, \forall y\,\big(x\,I\,y \rightarrow y\,I\,x\big)$ .

The strategy: to prove an implication, assume the antecedent (rule CP), unravel the definition with D1, swap the conjuncts with a tautology, re-fold the definition, discharge the conditional premise, then universally generalize.

\begin{aligned} &\text{Goal: } \forall x\, \forall y\,\big(x\,I\,y \rightarrow y\,I\,x\big) \\ 1.\quad & x\,I\,y && \text{P (assume antecedent; } x,y \text{ flagged)} \\ 2.\quad & xQy \land yQx && \text{1, D1} \\ 3.\quad & yQx \land xQy && \text{2, T } (\alpha \land \beta \equiv \beta \land \alpha) \\ 4.\quad & y\,I\,x && \text{3, D1} \\ 5.\quad & x\,I\,y \rightarrow y\,I\,x && \text{CP, 1–4 (premise discharged)} \\ 6.\quad & \forall x\, \forall y\,\big(x\,I\,y \rightarrow y\,I\,x\big) && \text{UG, 5} \end{aligned}

At line 5 the conditional premise is discharged, so $x$ and $y$ are no longer flagged — only then is the generalization at line 6 legitimate.

Summary

The grammar of first-order logic sorts every symbol into connectives, terms (with operators), predicates/relations, and quantifiers; negating a quantifier flips it, and quantifier order is significant.
Universal Specification removes a $\forall$ by substituting any one term consistently for the quantified variable; Universal Generalization restores a $\forall$ , but never on a flagged variable.
A flagged variable is free and introduced in a premise; generalizing on one before discharging that premise lets you prove falsehoods.
Quantifier proofs follow four steps: symbolize, drop quantifiers (US), reason with P/CP/T/RAA, re-quantify (UG).
An axiomatic system has undefined terms, axioms, definitions, and theorems; the weak-preference system rests on one undefined relation $Q$ with transitivity (A1) and totality (A2), plus indifference (D1) and strong preference (D2).
Working in such a system means reasoning only from the axioms and definitions — resisting the urge to import outside intuition about what the symbols "really" mean.